Berry–Esseen Inequalities: Convergence Bounds

Updated 10 December 2025

Berry–Esseen type inequalities are explicit bounds that measure the difference between a normalized sum’s distribution and the Gaussian distribution in terms of moment conditions.
They have been refined for various settings including i.i.d. variables, dependency graphs, and multivariate frameworks to optimize convergence in central limit theorems.
These inequalities support practical applications in statistical estimation, probability theory, and computational techniques by providing sharp bounds for normal approximations.

A Berry-Esseen type inequality precisely quantifies the rate of convergence in the central limit theorem (CLT), bounding the difference between the cumulative distribution function (cdf) of a normalized sum of independent random variables and the Gaussian cdf. These inequalities provide explicit rates under moment and dependence conditions, and have been refined for a variety of settings: classic i.i.d., dependent graphs, weak dependence, random walks conditionally constrained, multivariate statistics, local limit theorems, functional laws, and even free probability.

The canonical Berry-Esseen theorem provides an explicit upper bound on the Kolmogorov distance $\sup_x |F_n(x) - \Phi(x)|$ between the distribution function $F_n$ of a normalized sum $S_n = (X_1+\dots+X_n)/\sqrt{n}$ of i.i.d., mean-zero, unit-variance random variables and the standard normal cdf $\Phi$ , in terms of the normalized third absolute moment: $\Delta_n = \sup_{x\in\mathbb R} | F_n(x) - \Phi(x) | \leq C_0 \frac{ \mathbb{E}|X_1|^3 }{ \sqrt{n} }$ Recent work has focused on reducing the absolute constant $C_0$ , with the currently best-known values $C_0 < 0.4748$ for i.i.d. (Shevtsova, 2011) and $C_0 < 0.5600$ for non-identical summands. Additionally, Zolotukhin et al. demonstrate that for i.i.d. Bernoulli variables, the optimal $C_0$ is at most $0.06\%$ above Esseen's lower bound $C_E \approx 0.4097$ for $n \leq 500{,}000$ (Zolotukhin et al., 2018).

For smooth test functions, Mattner–Shevtsova established refined inequalities in Zolotarev $\zeta_3$ -distance: $\zeta_3( \ast_{i=1}^n P_i, \ast_{i=1}^n Q_i ) \leq (1/6 \sigma^3) \sum_i \sigma_i^3 B(\rho_i)$ where $Q_i$ are symmetric two-point laws, $\rho_i = \nu_3(P_i)/\sigma_i^3$ , and $B(\rho)$ is explicitly smaller than $\rho$ for $\rho>1$ , strictly improving Tyurin's optimal bounds (Mattner et al., 2017).

2. Optimality, Nonuniform Inequalities, and Smoothing Techniques

The uniform Berry–Esseen bound does not reflect tail improvements. Nonuniform Berry–Esseen inequalities, due to Nagaev and refined by Pinelis, provide bounds of the form: $| \mathbb{P}( S > B x ) - \mathbb{P}( Z > x ) | \leq C_{nu} \frac{A}{B^3 (1 + x^3) }$ for normalized sums $S$ with third moment $A = \sum \mathbb{E}|X_i|^3$ (Pinelis, 2013, Pinelis, 2013). Innovations in Fourier-based smoothing inequalities and truncation/exponential tilt permit sharper tail behavior and constants. Pinelis shows $C_{nu} \leq 4.5$ in the i.i.d. regime, considerably narrowing the gap from earlier $\sim$ 25x factors.

3. Berry–Esseen Bounds under Dependence Structures

Several extensions generalize Berry–Esseen type bounds to various dependence frameworks:

Dependency graphs: For sums $S_n$ of random variables $Y_i$ linked by a sparse dependency graph of degree $\Delta$ , Fourier-analytic techniques yield Kolmogorov bounds matching the independence rate (up to $n/(1+\Delta)$ replacement), improving over earlier Stein-method-based constants (Janisch et al., 2022).
Weak dependence (Bernoulli-shift): With stationary sequences admitting sufficiently summable coupling coefficients, the Berry–Esseen rate is $n^{p/2-1}$ for $p\in (2,3]$ , optimal for time series models, dynamical systems, and nonlinear recursions (Jirak, 2016).
Self-normalization/local dependence: Stein’s method and concentration inequalities allow optimal ( $n^{-1/2}$ ) uniform rates for self-normalized statistics under mild local dependence and only third-moment conditions (Zhang, 2021).

4. Multidimensional, Nonlinear, and Conditional Extensions

Multivariate CLTs: The quantification of convergence rate in the multidimensional central limit theorem involves intricate combinatorial smoothing (partition–Möbius operator), yielding Berry–Esseen bounds of order $O(n^{-1/2})$ modulo combinatorial constants depending on dimension (Heuberger et al., 2016).
General nonlinear statistics: Using randomized multivariate concentration, Berry–Esseen-type rates of $O(d^{1/2} n^{-1/2})$ are established for wide classes of estimators, including M-estimators and stochastic gradient descent averages, with explicit dependence on third moment and nonlinearity envelope (Shao et al., 2021).
Conditional sums: For sums of independent random variables conditioned on another sum (e.g., occupancy, hashing, branching), a Berry–Esseen inequality with $O(N^{-1/2})$ rate holds under moment and smoothing assumptions (Klein et al., 2019).

5. Berry–Esseen for Random Processes and Functional Laws

Local limit theorems in $\mathbb{R}^d$ : Explicit Berry–Esseen bounds in terms of Lyapunov coefficients $L_3$ , density maxima $M$ , and dimension $d$ are achieved for the uniform difference between convolution densities and the Gaussian, allowing control even for log-concave measures and symmetric cases (Bobkov et al., 30 Jul 2024).
Breuer–Major functional CLT: For sums $V_n$ of functionals $\varphi(X_k)$ of stationary Gaussian sequences, Malliavin–Stein plus Gebelein's maximal correlation yield explicit bounds in total variation, optimal in $n^{-1/2}$ rate under $\ell^1$ covariances and Hermite rank assumptions (Nourdin et al., 2018).
Fast convergence under density assumptions: If i.i.d. summands match moments to order $k$ and have a small density piece, the Kolmogorov distance achieves $O(n^{-(k-1)/2})$ rates plus exponentially small residual, sharply improving upon the classical $n^{-1/2}$ (Johnston, 2023).
Berry–Esseen for conditioned random walks: For random walks conditioned to stay positive, the convergence to Rayleigh limit law admits $O(n^{-1/2})$ Berry–Esseen bounds, with all constants explicit and third-moment dependence cubic in the denominator (Denisov et al., 11 Dec 2024).
Fractional Ornstein-Uhlenbeck and Gaussian-driven models: For ergodic (moment) estimators of drift under discrete observation, Berry–Esseen bounds $C_{\theta,H,h} n^{-1/2}$ hold for Hurst parameter $H \le 5/8$ , and $C_{\theta,H,h} n^{4H-3}$ for $H > 5/8$ , uniformly in fixed mesh $h$ , even extending to various general Gaussian drivers (Tang et al., 3 Apr 2025).

6. Applications, Optimization, and Computational Techniques

Berry–Esseen bounds have critical impact in:

Small deviation and probability inequalities: By combining Berry–Esseen bounds with moment semidefinite programming, new sharp lower bounds for small-deviation probabilities (Feige’s conjecture) are established (Guo et al., 2020).
Sequential decoding complexity: Berry–Esseen bounds enable uniform upper estimates for decoding complexity in communications (GDA, MLSDA algorithms), valid for arbitrary blocklength and all SNR regimes [0701026].

Computational methods—gridding, interpolation, and local mass approximations—are essential for sharp constant estimation, particularly for lattice laws and Bernoulli variables, achieving global bounds up to $0.06\%$ above theoretical minimums (Zolotukhin et al., 2018).

7. Extensions to Free Probability

In noncommutative probability, Berry–Esseen-type inequalities for freely independent $X_i$ with only fourth moments assume explicit rates (Kolmogorov distance to semicircle law) $O(n^{-1/2+\varepsilon})$ , nearly matching the commutative case (Neufeld, 30 Mar 2025). Subordination of F-transforms facilitates this analysis, and further research aims to remove the $\varepsilon$ -loss under minimal moment conditions.

In summary, Berry–Esseen type inequalities provide optimal and explicit quantitative bounds for normal approximation under a spectrum of independence, dependence, moment, and structural assumptions. Advances include dimension-explicit multivariate results, nonuniform inequalities suited for tail regimes, computational optimization of constants, improved rates under symmetry/density, extensions to functional and free-probability contexts, and powerful tools for modern statistical estimation and probability theory.